Prior to death, I tried to cast Amumu's ultimate. I assumed my spell did not cast because I did not see any cooldown displayed. In the next immediate team fight (ARAM game), his Ultimate failed to cast.
It is unclear to me whether or not the Ultimate casted successfully before death. 
This happens with any champion. I've had it happen with Malzahar most recently. It occurs when you initiate a command (to cast the spell) and die before that command is registered by the servers. That's like a 10th of a second window immediately before your death.
Anyway, when this occurs your client registers your death and won't perform the full ult. The server registers that you have used your ability and puts it on cooldown but your client has become out of sync due to a race condition in Riot's ability networking code. Basically, if you hear/see your ult or another ability cast but die immediately after and the cooldown is not running wait the labeled cooldown (you'll have to calculate this manually), after the normal cooldown has run you should be all set and everything will function correctly again.
It's a bug, but I imagine it's low priority as it will have significant risk requiring the reworking of a lot of the ability/action networking code. Whichever Architect designed the system originally either didn't do a good enough job or the developer overlooked some implementation detail. If it is the later and not the former the fix should be easier.
